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Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices
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Proves BBP-type phase transition fluctuations for the largest eigenvalue of entrywise-transformed spiked Wigner matrices: Gaussian fluctuations (order N^{-1/2}) above the effective-SNR threshold and GOE Tracy-Widom (N^{-2/3}) below it, via Green function comparison.
Problem
Fluctuations of the largest eigenvalue of transformed (entrywise-function-applied) spiked Wigner matrices were unknown, though they matter for detection with improved PCA and for finite-N error probabilities.
Approach
The entrywise transform acts as a rank-2 spiked Wigner-type perturbation whose noise is non-Wigner; the authors interpolate a Wigner-type matrix flow and use local laws plus Green function comparison to transfer known spiked Wigner fluctuation results to the transformed matrix, with effective SNR lambda_e = lambda (E f'(sqrt(N)W_12))^2.
Key results
- For N=1024 with lambda=0.8 (lambda_e ≈ 2.902) and bimodal noise, the rescaled largest eigenvalue fluctuations match the predicted Gaussian limit with variance 2(lambda_e-1)/lambda_e.
- For N=1024 with lambda=0.1 (lambda_e ≈ 0.363), N^{2/3}(mu_1 - 2) matches the GOE Tracy-Widom law.
- Rigidity: supercritical mu_1 - (sqrt(lambda_e)+1/sqrt(lambda_e)) = O(N^{-1/2+eps}); subcritical mu_1 - 2 = O(N^{-2/3+eps}) with overwhelming probability.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply the optimal entrywise transformation to non-Gaussian bimodal-noise data | Spiked Wigner PCA · Non-Gaussian bimodal noise, λ = 0.8 | effective SNR: 0.8 -> \lambda_e \approx 2.902 | Helped |
| Apply the optimal entrywise transformation to non-Gaussian bimodal-noise data | Spiked Wigner PCA · Non-Gaussian bimodal noise, λ = 0.1 | effective SNR: 0.1 -> \lambda_e \approx 0.363 | Helped |
| Apply the polynomial entrywise transformation f(x) = (x^{2}+3x-1)/\sqrt{11} | Spiked Wigner PCA with Gaussian noise · Gaussian noise, λ = 2.5 | effective SNR: 2.5 -> \lambda_e \approx 1.294 | Hurt |
| Apply the polynomial entrywise transformation f(x) = (x^{2}+3x-1)/\sqrt{11} | Spiked Wigner PCA with Gaussian noise · Gaussian noise, λ = 0.1 | effective SNR: 0.1 -> \lambda_e \approx 0.350 | Helped |
| Apply entrywise transformation f with effective SNR above the critical value | Transformed spiked Wigner · Supercritical regime (λ_e > 1) | largest-eigenvalue fluctuation law | Helped |
| Apply entrywise transformation f with effective SNR below the critical value | Transformed spiked Wigner · Subcritical regime (λ_e < 1) | largest-eigenvalue fluctuation law | Helped |
Limitations
- Requires technical assumptions on the spike (bounded max entry, sums of moments) and on noise moments; the authors say these may be weakened.
- Analysis restricted to rank-1 spikes; rank-k case only conjectured.
- Proposition bounds (e.g., subcritical difference N^{-3/4}) are not optimal per the authors.
Experimental setup
Metrics: histogram agreement with limiting Gaussian and GOE Tracy-Wid rigidity of largest eigenvalue
Baselines: spiked Wigner matrix (untransformed PCA) transformed PCA with optimal f = -p'/p
Builds on: Baik-Ben Arous-Péché transition Perry-Wein-Bandeira-Moitra transformed PCA Green function comparison (Erdős-Yau-Yin)
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